Evidence

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The entry point checks the statement of the owning account: the rotated deficit integral is below $-1$ on every circle and line of the pencils of $z^4-\tfrac32z^2-z$ and $z^4-2z^2$. From the repository root run:

uv run --no-sync python wiki/research/erdos_114/quartic_evidence/evidence/main.py

It uses the repository's tools.Checker, python-flint arb and acb balls at 80 bits, and numpy only for approximate roots that Rouché tests then certify. It reads no input files and writes no output. The full check covers the tangent-chord angle range $[0,\pi/2]$ of both pencils. It takes about forty-five minutes of one thread for each stratum, with a peak resident memory near 51 MB. --only double or --only equal restricts the check to one stratum, so the two strata can run as separate processes. --quick checks four boxes of each pencil and the controls in a few seconds.

The program prints, for each stratum, the number of boxes and pieces and the largest upper bound over its boxes. Every box must have an upper bound below $-1$; a box that fails is bisected, at most six times, and the boxes must tile the range. Controls on the box $[0,10^{-4}]$, which contains the line, check that the upper and lower bounds are ordered and less than $1/2$ apart, and that they enclose the value of the integral on the line computed separately by floating-point quadrature, $-1.7075$ for the double critical point and $-2.3963$ for equal values. The second is a negative control: no target below that value can pass on that box. Tolerances affect only how many pieces are used. A failed certificate, a nonfinite enclosure or an inverted piece makes the box fail rather than pass.

The landscape probe, enrolled as an entry point, recomputes in floating point the exploratory observations of the sketch in a few seconds, including the identities (14) and (15) at sample shapes, the critical equilateral shape, the cubic analogue and the convexity of the length in the gaps between the critical values, also in degrees five and six. It also compares the full second-variation formulas of L178 with finite differences at two shapes. For the angular-gap Hessian the step is at most one fiftieth of the smallest gap and the error is scaled by the Hessian's largest entry; derivatives in these coordinates can be large near a vertex. Further checks reproduce the failed sharp moment comparison, the unequal petal lengths and variances at the equilateral shape, and the negative deficits observed after replacing the density by twice the Euclidean norm of the inverse derivatives, at six regular shapes and sixty angles on each collision pencil. None of these samples covers a parameter region. Run it with:

uv run --no-sync python wiki/research/erdos_114/quartic_evidence/evidence/landscape.py

Its checks only confirm that the recorded observations still come out of the stated double-precision computation; they bound nothing.

The program does not check identity (10), the parametrization of the strata by the pencils, or the estimates (a) to (d) of the quartic sketch, which proves them. This is author evidence, not an independent examination.